Introduction to
AC/DC Module
Version 4.3b
Introduction to the AC/DC Module
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Version:
May 2013
COMSOL 4.3b
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Part No. CM020104
Contents
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
The Use of the AC/DC Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
The AC/DC Module User Interfaces . . . . . . . . . . . . . . . . . . . . . . 6
The Physics List by Space Dimension and Study Type . . . . . . . . . 11
The Model Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Tutorial Example: Modeling a 3D Inductor. . . . . . . . . . . . . . . . . 14
|i
ii |
Introduction
The AC/DC Module is used by engineers and scientists to understand, predict,
and design electric and magnetic fields in static, low-frequency and transient
applications. Simulations of this kind result in more powerful and efficient
products and engineering methods. It allows its users to quickly and accurately
predict electromagnetic field distributions, electromagnetic forces, and power
dissipation in a proposed design. Compared to traditional prototyping, COMSOL
helps to lower costs and can evaluate and predict entities that are not directly
measurable in experiments. It also allows the exploration of operating conditions
that would destroy a real prototype or be hazardous.
The AC/DC Module includes stationary and dynamic electric and magnetic fields
in two-dimensional and three-dimensional spaces along with traditional
circuit-based modeling of passive and active devices. All modeling formulations
are based on Maxwell’s equations or subsets and special cases of these together
with material laws like Ohm’s law for charge transport. The modeling capabilities
are accessed via a number of predefined user interfaces, referred to as AC/DC user
interfaces, which allow you to set up and solve electromagnetic models. The 
AC/DC user interfaces cover electrostatics, DC current flow, magnetostatics, AC
and transient current flow, AC and transient magnetodynamics, and AC
electromagnetic (full Maxwell) formulations.
Under the hood, the AC/DC user interfaces formulate and solve the differential
form of Maxwell’s equations together with initial and boundary conditions. The
equations are solved using the finite element method with numerically stable edge
element discretization in combination with state-of-the-art algorithms for
preconditioning and solution of the resulting sparse equation systems. The results
are presented in the graphics window through predefined plots of electric and
magnetic fields, currents and voltages or as expressions of the physical quantities
that you can define freely, and derived tabulated quantities (for example resistance,
capacitance, inductance, electromagnetic force, and torque) obtained from a
simulation.
The work flow in the module is straightforward and is described by the following
steps: define the geometry, select materials, select a suitable AC/DC user
interface, define boundary and initial conditions, define the finite element mesh,
select a solver, and visualize the results. All these steps are accessed from the
COMSOL Desktop. The solver selection step is usually carried out automatically
using the default settings, which are already tuned for each specific AC/DC user
interface.
The AC/DC Module Model Library describes the available features using tutorial
and benchmark examples for the different formulations. The library contains
Introduction
|1
models for industrial equipment and devices, tutorial models, and benchmark
models for verification and validation of the AC/DC user interfaces. See “The
Model Library” on page 13 to start exploring now.
This guide will give you a jump start to your modeling work. It has examples of
the typical use of the module, a list of all the AC/DC user interfaces including a
short description for each, and a tutorial example that introduces the modeling
workflow.
The Use of the AC/DC Module
The AC/DC Module can model electric, magnetic, and electromagnetic fields in
statics and low-frequency applications. The latter means that it covers the
modeling of devices that are up to about 0.1 electromagnetic wavelengths in size.
Thus, it can be used to model nanodevices up to optical frequencies or human size
devices operated at frequencies up to 10 MHz.
AC/DC simulations are frequently used to extract circuit parameters. Figure 1 on
the next page shows the electric potential and the magnetic flux lines for a
micro-scale square inductor, used for LC bandpass filters in
microelectromechanical systems (MEMS). A DC voltage is applied across the
electrically conducting square shaped spiral, resulting in an electric current flow
that in turn generates a magnetic field through and around the device.
The distribution and strength of electric and magnetic fields arising from applied
current and voltage can be translated into resistance, capacitance, and inductance
values for an equivalent circuit model that is easy to understand and analyze from
both qualitative and quantitative perspectives. This kind of understanding in terms
of a simplified model is usually the first step when creating or improving a design.
Another common use of this module is to predict forces and motion in electric
motors and actuators of a wide range of scales.
2 | Introduction
Figure 1: Electric potential distribution and magnetic flux lines for a MEMS inductor.
In Figure 2, Figure 3, and Figure 4, the dynamics of a magnetic brake is shown.
The brake consists of a disc of conductive material and a permanent magnet. The
magnet generates a constant magnetic field, in which the disc is rotating. When a
conductor moves in a magnetic field it induces currents, and the Lorentz forces
from the currents counteracts the spinning of the disc.
The braking torque is computed from the magnetic flux and eddy current
distributions. It is then fed into a rigid body model for the dynamics of the
spinning disc that is solved simultaneously with the electromagnetic model.
Introduction
|3
Figure 2: The eddy current magnitude and direction in the spinning disc of a magnetic brake.
Figure 3: The time evolution of the braking torque.
4 | Introduction
Figure 4: The time evolution of the angular velocity.
The AC/DC Module includes a range of tools to evaluate and export results, for
example, the evaluation of force, torque, and lumped circuit parameters like
resistance, capacitance, inductance, impedance, and scattering matrices
(S-parameters). Lumped parameters can also be exported in the Touchstone file
format.
The combination of fully distributed electromagnetic field modeling and
simplified circuit based modeling is an ideal basis for design, exploration, and
optimization. More complex system models can be exploited using circuit-based
modeling while maintaining links to full field models for key devices in the circuit
allowing for design innovation and optimization on both levels.
Introduction
|5
The AC/DC Module User Interfaces
The AC/DC physics user interfaces are based upon Maxwell’s equations or
subsets and special cases of these together with material laws. Different subsets of
Maxwell’s equations in combination with special material laws result in different
electric, magnetic, or electromagnetic formulations. In the module, these laws of
physics are translated by the AC/DC user interfaces to sets of partial differential
equations with corresponding initial and boundary conditions.
An AC/DC user interface defines a number of features. Each feature represents a
term or condition in the underlying Maxwell-based formulation and may be
defined in a geometric entity of the model, such as a domain, boundary, edge (for
3D models), or point.
Figure 5 on the next page uses the Capacitor Tunable model (found in the AC/
DC Module Model Library) to show the Model Builder window and the settings
window for the selected Charge Conservation 1 feature node. The Charge
Conservation 1 node adds the terms representing Electrostatics to the model
equations for a selected geometrical domain in the model.
Furthermore, the Charge Conservation 1 node may link to the Materials node to
obtain physical properties, in this case the relative permittivity of a user-defined
dielectric. The properties, defined by the Dielectric material node, may be
functions of the modeled physical quantities, such as temperature. The Zero
6 | The AC/DC Module User Interfaces
Charge 1 boundary condition feature adds the natural boundary conditions that
limit the Electrostatics domain.
Figure 5: The Model Builder window (to the left), and the Charge Conservation 1 settings window for the
selected feature node (to the right). The Equation section in the settings window shows the model
equations and the terms added to the model equations by the Charge Conservation 1 node. The added
terms are underlined with a dotted line. The text also explains the link between the material properties
represented by the Dielectric node and the values for the Relative permittivity.
The AC/DC Module User Interfaces
|7
The AC/DC Module has several AC/DC user interfaces ( ) for different types
of electric and magnetic modeling. Figure 6 shows the AC/DC user interfaces as
well as those under the Heat Transfer branch, which contains the Joule Heating
and Induction Heating user interfaces. Also see “The Physics List by Space
Dimension and Study Type” on page 11. A brief overview of the AC/DC user
interfaces follows.
Figure 6: The AC/DC Module physics as displayed in the Model Wizard.
E LECTRIC C URRENTS
The Electric Currents user interface ( ) is used to model DC, AC, and transient
electric current flow in conductive and capacitive media. This user interface solves
a current conservation equation for the electric potential. Some examples of its use
are designing busbars for DC current distribution and designing AC capacitors.
E LECTRIC C URRENTS ,
SHELL
The Electric Currents, Shell user interface ( ) is available in 2D, 2D
axisymmetric, and 3D geometries. It is used to model DC, AC, and transient
electric current flow confined to conductive and capacitive thin
current-conducting shells of fixed or varying thickness. This user interface solves
a boundary current conservation equation for the electric potential. Modeling
ground return current flow in the hull of a ship or in the body of a car are examples
of simulations that can be done with this user interface.
8 | The AC/DC Module User Interfaces
E LECTRICAL C IRCUIT
The Electrical Circuit user interface ( ) has the equations to model electrical
circuits with or without connections to a distributed fields model, solving for the
voltages, currents, and charges associated with the circuit elements. Circuit models
can contain passive elements like resistors, capacitors, and inductors as well as
active elements such as diodes and transistors.
E LECTROSTATICS
The Electrostatics user interface ( ) solves a charge conservation equation for
the electric potential given the spatial distribution of the electric charge. It is used
primarily to model charge conservation in dielectrics under static conditions. This
user interface applies also under transient conditions but then it is usually
combined with a separate transport model for single species or multispecies charge
transport. Such transport models can be found in the Chemical Reaction
Engineering Module and in the Plasma Module. AC simulations are supported
and used especially together with the piezoelectric user interface in the Acoustics,
Structural Mechanics and MEMS Modules. Some typical applications that are
simulated in the Electrostatics user interface are capacitors, dielectric sensors, and
bushings for high voltage DC insulation.
M AGNETIC F IELD FORMULATION
The Magnetic Field Formulation user interface ( ) solves Faraday’s law for the
magnetic H-field. It is used to model mainly AC, and transient magnetodynamics
in conducting domains. It is especially suitable for modeling involving nonlinear
conductivity effects, for example in superconductors.
M AGNETIC F IELDS
The Magnetic Fields user interface ( ) solves Ampère’s law for the magnetic
vector potential. It is used to model magnetostatics, AC, and transient
magnetodynamics. Magnets, magnetic actuators, electric motors, transformers,
induction based nondestructive testing, and eddy current generation are typical
applications for this user interface. It supports both linear media and media with
magnetic saturation.
M AGNETIC F IELDS , N O C URRENTS
The Magnetic Fields, No Currents user interface ( ) is used to efficiently model
magnetostatics in current free regions, for example, when designing permanent
magnet-based devices. It solves a magnetic flux conservation equation for the
magnetic scalar potential. This user interface supports both linear media and
media with magnetic saturation.
The AC/DC Module User Interfaces
|9
M AGNETIC
AND
E LECTRIC F IELDS
The Magnetic and Electric Fields user interface ( ) is used to model
magnetostatics and AC magnetodynamics. It solves Ampère’s law for the magnetic
vector potential together with a current conservation equation for the electric
potential. The application areas are mostly the same as for the Magnetic Fields user
interface. The addition of the electric potential makes it more flexible as also
electric boundary conditions are available.
R OTATING M ACHINERY, M AGNETIC
The Rotating Machinery, Magnetic user interface ( ) combines the magnetic
fields (magnetic vector potential) and magnetic fields, no currents (scalar magnetic
potential) formulations with a selection of predefined frames for prescribed
rotation or rotation velocity—it shares most of its features with the Magnetic
Fields user interface. This user interface requires that the geometry is created as an
assembly from individual parts for the rotor and stator.
I NDUCTION H EATING
The Induction Heating multiphysics user interface ( ) combines all features
from the Magnetic Fields user interface in the time-harmonic formulation with the
Heat Transfer user interface to model induction and eddy current heating. The
predefined interaction adds the electromagnetic losses from the magnetic fields as
a heat source. This user interface is based on the assumption that the magnetic
cycle time is short compared to the thermal time scale (adiabatic assumption).
JOULE H EATING
The Joule Heating multiphysics user interface ( ) combines all features from the
Electric Currents user interface with the Heat Transfer user interface to model
resistive heating and heating due to dielectric losses. The predefined interaction
adds the electromagnetic losses from the electric currents as a heat source.
10 | The AC/DC Module User Interfaces
The Physics List by Space Dimension and Study Type
The table list the physics user interfaces available specifically with this module in
addition to the COMSOL Multiphysics basic license.
PHYSICS
ICON
TAG
SPACE DIMENSION
PRESET STUDY TYPE
Electric Currents*
ec
all dimensions
stationary; frequency
domain; time dependent;
small signal analysis,
frequency domain
Electric Currents - Shell
ecs
3D, 2D, 2D
axisymmetric
stationary; frequency
domain; time dependent;
small signal analysis,
frequency domain
Electrical Circuit
cir
Not space
dependent
stationary; frequency
domain; time dependent
Electrostatics*
es
all dimensions
stationary; time dependent;
frequency domain; small
signal analysis, frequency
domain
Magnetic Field Formulation
mf
3D, 2D, 2D
axisymmetric
stationary; frequency
domain; time dependent;
small signal analysis,
frequency domain
Magnetic Fields*
mf
3D, 2D, 2D
axisymmetric
stationary; frequency
domain; time dependent;
small signal analysis,
frequency domain; coil
current calculation
Magnetic Fields, No Currents
mfn
c
3D, 2D, 2D
axisymmetric
stationary; time dependent
Magnetic and Electric Fields
mef
3D, 2D, 2D
axisymmetric
stationary; frequency
domain; small signal
analysis, frequency domain
Rotating Machinery, Magnetic
rm
m
3D, 2D
stationary; time dependent;
coil current calculation
AC/DC
The AC/DC Module User Interfaces
| 11
PHYSICS
ICON
TAG
SPACE DIMENSION
PRESET STUDY TYPE
Induction Heating
ih
3D, 2D, 2D
axisymmetric
stationary; frequency
domain; time dependent;
frequency stationary;
frequency transient
Joule Heating*
jh
all dimensions
stationary; frequency
domain; time dependent;
frequency stationary;
frequency transient
Electromagnetic Heating
* This is an enhanced user interface, which is included with the base COMSOL package but
has added functionality for this module.
12 | The AC/DC Module User Interfaces
The Model Library
To open an AC/DC Module Model Library model, select View > Model Library
from the main menu in COMSOL Multiphysics. In the Model Library
window that opens, expand the AC/DC Module folder and browse or search the
contents. Click Open Model and PDF to open the model in COMSOL
Multiphysics and a PDF to read background theory about the model including the
step-by-step instructions to build it. The MPH-files in the COMSOL model
libraries can have two formats—Full MPH-files or Compact MPH-files.
• Full MPH-files, including all meshes and solutions. In the Model Library
these models appear with the
icon. If the MPH-file’s size exceeds 25MB,
a tip with the text “Large file” and the file size appears when you position
the cursor at the model’s node in the Model Library tree.
• Compact MPH-files with all settings for the model but without built meshes
and solution data to save space on the DVD (a few MPH-files have no
solutions for other reasons). You can open these models to study the settings
and to mesh and re-solve the models. It is also possible to download the full
versions—with meshes and solutions—of most of these models through
Model Library Update. These models appear in the Model Library with the
icon. If you position the cursor at a compact model in the Model Library
window, a No solutions stored message appears. If a full MPH-file is available
for download, the corresponding node’s context menu includes a Download
Full Model item (
).
The next section uses a model from the Model Library. Go to “Tutorial Example:
Modeling a 3D Inductor” to get started.
The Model Library
| 13
Tutorial Example: Modeling a 3D Inductor
Inductors are used in many applications for low-pass filtering or for impedance
matching of predominantly capacitive loads. They are used in a wide frequency
range from near static up to several MHz. An inductor usually has a magnetic core
to increase the inductance while keeping its size small. The magnetic core also
reduces the electromagnetic interference with other devices as the magnetic flux
tends to stay within it. Because there are only crude analytical or empirical
formulas available to calculate impedances, computer simulations or
measurements are necessary during the design stage. In general, inductor
modeling is more complex than modeling resistors and capacitors but similar
principles apply. Using an external CAD software to design and draw the model,
the geometry is imported into the AC/DC Module for static and frequency
domain analysis. The inductor geometry is shown in Figure 7.
Core
Cu winding
96 mm
Feed gap
Figure 7: The inductor geometry.
14 | Tutorial Example: Modeling a 3D Inductor
First a magnetostatic simulation is performed to get the DC inductance. At low
frequencies capacitive effects are negligible. A relevant equivalent circuit model is
an ideal inductor in a series with an ideal resistor. The inductance and the
resistance are both computed in the magnetostatic simulation. At a high
frequency, capacitive effects become significant. The equivalent circuit model
involves connecting an ideal capacitor in parallel with the DC circuit. The circuit
parameters are obtained by analyzing the frequency dependent impedance
obtained from a frequency domain simulation. In this tutorial, the AC analysis is
done up to the point when the frequency dependent impedance is computed.
These step-by-step instructions guide you through the detailed modeling of an
inductor in 3D. The module also has a physics user interface to model electrical
circuits, which is detailed in the magnetostatic part of this model. The first step is
to perform a DC simulation.
M o d e l W i z a rd
1 Double-click the COMSOL Multiphysics icon on the desktop.
2 In the Model Wizard, the space dimension defaults to 3D. Click Next
3 In the Add Physics tree under AC/DC, select Magnetic Fields (mf)
4 Click Add Selected
then click Next
.
.
5 In the Studies tree, select Preset Studies>Stationary
6 Click Finish
.
.
.
Geo metr y 1
The main geometry is imported from a file. Air domains are typically not part of a
CAD geometry, so these are added to the model. For convenience, three
additional domains are defined in the CAD file and then used to define a narrow
feed gap where an excitation is applied.
Note: The location of the files used in this exercise varies based on the
installation. For example, if the installation is on your hard drive, the file path
might be similar to C:\Program Files\COMSOL43a\models\.
Tutorial Example: Modeling a 3D Inductor
| 15
Import 1
1 Under Model 1, right-click Geometry 1
and select Import .
2 In the settings window under Import,
click Browse. Then navigate to your
COMSOL Multiphysics installation
folder, locate the subfolder
ACDC_Module\Inductive_Devices_and
_Coils, select the file
inductor_3d.mphbin and click Open.
3 Click Import.
Sphere 1
1 In the Model Builder, right-click
Geometry 1
and choose Sphere
.
2 Go to the Sphere settings window. Under
Size and Shape in the Radius field, enter
0.2.
3 Click to expand the Layers section. In the
associated table, under Thickness enter
0.05.
4 Click the Build All
button.
Form Union
1 In the Model Builder under Geometry 1,
click Form Union .
2 In the Finalize settings window, click
Build All .
16 | Tutorial Example: Modeling a 3D Inductor
3 On the Graphics window toolbar, click the Zoom Extents
the Wireframe Rendering
button.
button and then
The geometry should match this figure.
Definitions - Selections
Use the Selection feature available under the Definitions node to make the set up
of materials and physics easier. Start by defining the domain group for the inductor
winding and continue by adding other useful selections. These steps illustrate how
to set up the Explicit nodes and rename the geometric selections accordingly.
1 In the Model Builder under Model 1, right-click Definitions
Selections>Explicit .
and choose
Tutorial Example: Modeling a 3D Inductor
| 17
2 Repeat Step 1 and add a total of six (6)
Explicit nodes . Or right-click the first
Explicit node and select Duplicate .
The following steps are done one at a time for each node using the table below.
3 In the Model Builder click an Explicit node
to open its settings window.
4 In the settings window under Input Entities, select Domain from the Geometric
entity level list. Use the table as a guide, and in the Graphics window, select the
domains as indicated. Then press F2 and rename the node.
DEFAULT NODE NAME
SELECT THESE DOMAINS
NEW NAME FOR THE NODE
Explicit 1
7, 8, and 14
Winding
Explicit 2
9
Gap
Explicit 3
6
Core
Explicit 4
1–4 and 10–13
Infinite Elements
Explicit 5
1–6 and 9–13
Non-conducting
Explicit 6
5,6, and 9
Non-conducting without
IE
There are many ways to select geometric entities. When you know the domain
and
to add, such as in this exercise, you can click the Paste Selection button
enter the information in the Selection field. In this example for the Explicit 1
node, enter 7,8,14 in the Paste Selection window. For more information
about selecting geometric entities in the Graphics window, see the COMSOL
Multiphysics Reference Manual.
18 | Tutorial Example: Modeling a 3D Inductor
5 After renaming all the nodes under Definitions, the sequence of nodes should
match this figure.
Definitions - In finite Elements
Use the Infinite Element feature available under the Definitions node to emulate
an infinite open space surrounding the inductor. The infinite element coordinate
scaling is recognized by any physics user interface supporting it - other physics user
interfaces are automatically disabled in the infinite element domains.
1 In the Model Builder under Model 1, right-click Definitions
Infinite Element Domain .
and choose
2 Go to the Infinite Elements settings window. Under Domain Selection select
Infinite Elements from the Selection list.
3 Under Geometry from the Type list, select Spherical.
Tutorial Example: Modeling a 3D Inductor
| 19
Mate rials
Now define the material settings for the different parts of the inductor. Use the
Selections defined in the previous section.
Copper
1 From the main menu, select
View>Material Browser .
2 Go to the Material Browser. In the
Materials tree under AC/DC, right-click
Copper and choose Add Material to
Model from the menu.
3 In the Model Builder, click Copper.
4 Go to the Material settings window.
Under Geometric Entity Selection select
Winding from the Selection list.
Air
1 Go to the Material Browser. In the
Materials tree under Built-In right-click
Air and choose Add Material to Model
from the menu.
2 In the Model Builder, click Air.
3 Go to the Material settings window.
Under Geometric Entity Selection select
Non-conducting from the Selection list.
20 | Tutorial Example: Modeling a 3D Inductor
Use r- Defined Mate rial 3
The core material is not included in the material library so it is entered as a
user-defined material.
1 In the Model Builder, right-click Materials
and choose Material .
2 Go to the Material settings window. Under
Geometric Entity Selection select Core
from the Selection list.
3 Under Material Contents in the table, enter these settings in the Value column:
- 0 for the Electrical conductivity
- 1 for the Relative permittivity
- 1e3 for the Relative permeability
4 Right-click Material 3 and choose Rename . In the Rename Material dialog
box and enter Core in the New name field.
5 Click OK. The node sequence under Materials should
match the figure to the right.
To view different labels such as the Identifier (mat1, mat2,
mat3) next to the material node name, select View>Model
Tutorial Example: Modeling a 3D Inductor
| 21
Builder Node Label and select one of the options. This changes what
combinations of labels are displayed in the Model Builder next to each node.
22 | Tutorial Example: Modeling a 3D Inductor
M ag n e t i c F i e l d s ( m f )
The model is solved everywhere except in the feed gap region. By keeping this
void, its default boundary conditions support surface currents closing the current
loop. In inductor modeling it is crucial to respect current conservation as this is a
law of nature (Ampère’s law).
1 In the Model Builder under Model 1, click Magnetic
Fields .
2 In the Magnetic Fields settings window, select
domains 1–8 and 10–14 only (all domains except 9).
Or first click the Clear Selection button , then click
the Paste Selection button and enter 1–8, 10–14 in
the Selection field.
Single-Turn Coil 1 and Boundary Feed 1
A dedicated feature is added for the solid copper winding of the inductor. This
feature makes it easier to excite the model in various ways.
1 Right-click Magnetic Fields and choose the domain setting Single-Turn 
Coil .
2 In the Single-Turn Coil settings window, under Domain Selection, choose
Winding from the Selection list.
3 Right-click Single-Turn Coil 1 and at the boundary level choose Boundary Feed
.
4 In the Boundary Feed settings window, to the right of the Boundary Selection
list, click the Clear Selection button .
5 Click the Paste Selection button
Click OK.
and enter 58 in the Selection field. 
Ground 1
1 In the Model Builder, right-click Single-Turn Coil 1 and choose the boundary
condition Ground .
2 In the Ground settings window, to the right of the Boundary Selection list, click
the Clear Selection button .
3 Click the Paste Selection button
Click OK.
and enter 79 in the Selection field. 
Tutorial Example: Modeling a 3D Inductor
| 23
The node sequence under Magnetic Fields should at
this point match the figure to the right.
Me sh 1
The steep radial scaling of the infinite elements region requires a swept mesh to
maintain a reasonably effective element quality.
Free Triangular 1 and Size
1 In the Model Builder under Model 1, right-click Mesh 1
Operations>Free Triangular .
and choose More
2 Select Boundaries 9–12, 68, 69, 73, and 76 only. Or click the Paste Selection
button
and enter 9–12, 68, 69, 73, 76 in the Paste Selection field.
3 In the Model Builder under Mesh 1, click Size . In the Size settings window
under Element Size, click the Predefined button. From the list select Coarser.
4 Click to expand the Element Size Parameters section. In the Minimum element
size field, enter 0.005.
5 In the Model Builder, right-click Free Triangular 1
Build Selected .
24 | Tutorial Example: Modeling a 3D Inductor
and select 
Swept 1 and Distribution 1
1 Right-click Mesh 1
and choose Swept
.
2 Go to the Swept settings window. Under Domain Selection:
- From the Geometric entity level list, select Domain.
- From the Selection list, select Infinite Elements.
3 Right-click Swept 1
Distribution .
and choose
4 Go to the Distribution settings window.
Under Distribution in the Number of
elements field, enter 4.
5 Click Build Selected
.
Free Tetrahedral 1
1 Right-click Mesh 1
Tetrahedral .
and choose Free
2 In the Model Builder, right-click Free
Tetrahedral 1
and choose Build
Selected .
The final node sequence under Mesh should match the figure.
Study 1
The magnetostatic model is now ready to solve.
1 In the Model Builder, right-click Study 1
and choose Compute
.
Tutorial Example: Modeling a 3D Inductor
| 25
Results
Magnetic Flux Density
The default plot group shows the magnetic flux density norm. At this point it is
important to consider whether the results make sense and to try and detect any
modeling errors.
26 | Tutorial Example: Modeling a 3D Inductor
Data Sets and Selections
By adding Selections and adjusting the Data Sets under the Results node you can
create more interesting plots.
1 In the Model Builder, expand the Results>Data Sets node. Right-click Solution
1
and choose Duplicate .
2 Right-click Solution 2
Selection .
and choose Add
3 Go to the Selection settings window.
Under Geometric entity selection:
- From the Geometric entity level list,
select Domain.
- From the Selection list, select Winding.
4 Right-click Solution 1
Duplicate .
and choose
5 Right-click Solution 3
and choose Add Selection
added to the Model Builder.
. A Selection node is
6 Go to the Selection settings window. Under Geometric Entity Selection:
- From the Geometric entity level list, select Domain.
- From the Selection list, select Core.
Tutorial Example: Modeling a 3D Inductor
| 27
3D Plot Group 2
1 In the Model Builder, right-click Results
2 Right-click 3D Plot Group 2
and choose 3D Plot Group
and choose Volume
.
.
3 Go to the Volume settings window. Under Data from the Data set list, select
Solution 2.
4 Click Replace Expression
in the upper-right corner of the Expression
section. From the menu, choose Magnetic Fields>Coil parameters>Coil
potential (mf.VCoil).
5 Under Coloring and Style from the Color
table list, select Thermal.
6 In the Model Builder, right-click 3D Plot
Group 2
and choose Volume . A
Volume 2 node is added to the Model
Builder.
7 Go to the Volume settings window.
Under Data from the Data set list, select
Solution 3.
28 | Tutorial Example: Modeling a 3D Inductor
8 Click the Plot
button.
On the Graphics window toolbar, click the Zoom In
button twice.
D e r i ved Val u e s
The coil inductance and resistance are available as predefined variables, which are
displayed as follows.
1 In the Model Builder under Results, right-click Derived Values
Global Evaluation .
and choose
Tutorial Example: Modeling a 3D Inductor
| 29
2 In the Global Evaluation settings window, locate the Expression section.
3 In the Expression edit field, enter mf.VCoil_1/mf.ICoil_1.
4 Click the Evaluate
button. The results are available in the Table window
located in the lower-right side of the COMSOL Desktop. Or select
View>Table
from the main menu to open the Table window.
5 In the Model Builder, right-click Derived Values
Evaluation .
and choose Global
6 In the Global Evaluation settings
window, in the Expression field, enter
2*mf.intWm/mf.ICoil_1^2.
7 Click the Evaluate
button and
compare the results in the Table
window.
You can also click the down arrow next to the Evaluate button on the settings
window to view each table. You should get about 0.29 m and 0.11 mH,
respectively. Below, both evaluations were made to the same table.
30 | Tutorial Example: Modeling a 3D Inductor
M ag n e t i c F i e l d s ( m f )
As can be seen in the Magnetic Flux Density plot, this is effectively contained
within the core. Thus, there should be little need to use infinite elements in this
model. Solve the model without the infinite elements to test this hypothesis.
1 In the Model Builder under Model 1,
click Magnetic Fields .
2 In the Magnetic Fields settings
window, select domains 5–8 and 14
only.
Study 1
1 In the Model Builder, right-click Study 1
and choose Compute
.
D e r i ved Val u e s
1 In the Model Builder under Results>Derived Values, click 
Global Evaluation 1 .
2 In the Global Evaluation settings window, click the Evaluate
Inductance value displays in the Table window.
3 In the Model Builder, click Global Evaluation 2
button. The
.
4 In the Global Evaluation settings window, click the Evaluate
button. The
Resistance value displays in the Table window. The results change very little.
Tutorial Example: Modeling a 3D Inductor
| 31
M o d e l W i z a rd
Next, add a simple circuit to the model.
1 In the Model Builder, right-click Model 1
and choose Add Physics
.
2 In the Add physics tree under AC/DC, double-click Electrical Circuit (cir)
to add it to the Selected physics list. You do not need to add any studies.
3 Click Finish
.
M ag n e t i c F i e l d s ( M F )
Change the field excitation so that it can connect to the circuit part of the model.
1 In the Model Builder, under Magnetic
Fields>Single-Turn Coil 1 click Boundary
Feed 1. .
2 In the Boundary Feed settings window under
Single-Turn Coil, choose Circuit (current) from
the Coil excitation list.
E l e c t r i c a l C i rcu it ( c i r )
Add a resistor in series with the inductor and drive. This is done for both by
applying a voltage source.
32 | Tutorial Example: Modeling a 3D Inductor
Voltage Source 1
1 In the Model Builder under Model 1,
right-click Electrical Circuit
and
choose Voltage Source .
The default settings correspond to a DC
source of 1 V between nodes 1and 0 in the
circuit net. Keep these settings.
Resistor 1
1 In the Model Builder, right-click Electrical
Circuit
and choose Resistor .
2 Go to the Resistor settings window.
Under Node Connections enter 1 and 2 in
the Node names table as in the figure to
the right.
3 Under Device Parameters in the R field,
enter 100[mohm].
Tutorial Example: Modeling a 3D Inductor
| 33
External I Vs. U 1
There is a special feature for connecting the circuit to the finite elements model.
1 In the Model Builder, right-click Electrical Circuit
External I Vs. U .
and choose 
2 In the External I Vs. U settings window
under Node Connections, enter Node
names as in to the figure to the right.
3 Under External Device, from the Electric
potential V list choose Coil voltage 
(mf/stcd1/stcdp1).
The sequence of nodes under Electrical
Circuit should match this figure.
Study 1
1 In the Model Builder, right-click Study 1
34 | Tutorial Example: Modeling a 3D Inductor
and choose Compute
.
Results
The current is now limited to approximately 10 A by the external resistor which
has a much larger resistance than that of the winding.
M o d e l W i z a rd - A dd a S e c o n d S t u d y
Now, set up the model for computing the frequency-dependent impedance.
1 In the Model Builder, right-click the top node Untitled.mph (root) and choose
Add Study .
Tutorial Example: Modeling a 3D Inductor
| 35
2 In the Select Study Type window that opens, under Studies>Preset Studies for
Selected Physics click Frequency Domain .
3 Find the Selected physics subsection.
4 In the Selected physics table under Solve for, click to change the check mark
to an
and remove the
Electrical Circuit (cir) user interface.
5 Click Finish
.
A Study 2 node with a Frequency Domain study step is added to the 
Model Builder.
36 | Tutorial Example: Modeling a 3D Inductor
Definitions
At high frequencies, the skin depth of the copper conductor in the winding
becomes difficult to resolve. The solution is to exclude the interior of the copper
domain from the simulation and instead represent the winding by a boundary
condition, which introduces surface losses via an equivalent surface impedance.
For this purpose add a boundary selection and an associated surface material.
Conductor Boundaries
1 In the Model Builder under Model 1,
right-click Definitions
and choose
Selections>Explicit .
2 Select Domains 7, 8, and 14 only.
3 Go to the Explicit settings window. Under
Output Entities from the Output entities
list, select Adjacent boundaries.
4 Press F2.
5 In the Rename Explicit window enter
Conductor Boundaries in the New name
field. Click OK.
Tutorial Example: Modeling a 3D Inductor
| 37
Mate rials
Copper (2)
1 From the main menu, select
View>Material Browser . In the
Materials tree under AC/DC, right-click
Copper and choose
Add to Current
Geometry.
2 In the Model Builder, click Copper (2).
3 Go to the Material settings window.
Under Geometric Entity Selection:
- From the Geometric entity level list,
select Boundary.
- From the Selection list, select
Conductor Boundaries.
M ag n e t i c F i e l d s ( m f )
1 In the Model Builder under Model 1, click Magnetic
Fields .
2 Go to the Magnetic Fields settings window. Under
Domain Selection from the Selection list, select
Non-conducting without IE.
38 | Tutorial Example: Modeling a 3D Inductor
Ampère's Law 2
Apart from the surface loss in the copper conductor, there are also AC losses in the
core. The loss in the core is introduced as an effective loss tangent. For this
purpose an extra equation/constitutive relation is required.
and choose Ampère's Law
1 In the Model Builder, right-click Magnetic Fields
. A second Ampère's Law node is added to the Model Builder.
2 Go to the Ampère's Law settings window. Under Domain Selection from the
Selection list, select Core.
3 Under Electric Field from the r list, select User defined and enter 1-5e-4*j in
the field.
Impedance Boundary Condition 1
1 In the Model Builder, right-click Magnetic Fields
choose Impedance Boundary Condition .
and at the boundary level,
2 Go to the Impedance Boundary Condition settings window. Under Boundary
Selection from the Selection list, select Conductor Boundaries.
Disable the Single-Turn Coil 1
The Single Turn Coil does not apply to an active domain anymore so it needs to
be disabled.
• In the Model Builder, under Magnetic Fields right-click Single-Turn Coil 1
and choose Disable
. The node is grayed out.
Tutorial Example: Modeling a 3D Inductor
| 39
Lumped Port 1
The electric potential is not available in this physics user interface, so to excite the
model a different boundary feature, which is more appropriate for high frequency
modeling, must be used.
1 In the Model Builder, right-click Magnetic Fields and at the boundary level
choose Lumped Port . A Lumped Port node is added to the sequence in the
Model Builder.
2 Go to the Lumped Port settings window.
Select boundaries 59–62 only.
The geometrical parameters of the
boundary set need to be entered manually.
3 Under Port Properties from the Type of
port list, select User defined.
- In the hport field, enter 0.024.
- In the wport field, enter 0.046.
4 Specify the ah vector as in the figure to 
the right.
5 From the Terminal type list, select Current.
The Model Builder node sequence under
Magnetic Fields should match this:
40 | Tutorial Example: Modeling a 3D Inductor
Study 2
Set up a frequency sweep from 1-10 MHz in steps of 1 MHz.
1 In the Model Builder, expand the Study 2 node and click Step 1: Frequency
Domain .
2 Under Study Settings, click the 
Range button .
3 In the Range window, enter these
settings:
- In the Start field, enter 1e6.
- In the Step field, enter 1e6.
- In the Stop field, enter 1e7.
4 Click Replace.
Near the resonance frequency, the problem becomes ill-conditioned. If there were
no loss, it would even become singular as the field solution then would approach
infinity. Thus for a high Q factor, the iterative solver may fail to converge and then
a direct solver must be used. Here it is sufficient to tweak the iterative solver to use
a more robust preconditioner.
Solver 2
1 In the Model Builder, right-click Study 2
.
and choose Show Default Solver
2 In the Model Builder, expand the
Study 2
>Solver Configurations
nodes as in the figure to the right and
then click the Iterative 1
node.
3 Go to the Iterative settings window.
Under General select Right from the
Preconditioning list.
4 In the Model Builder, right-click 
Study 2
and choose Compute .
Tutorial Example: Modeling a 3D Inductor
| 41
Results
Magnetic Flux Density
Again check the default plot for any modeling errors. Note that the magnetic flux
density inside the copper winding is not computed as this domain was excluded.
Plot the surface current distribution at the lowest frequency solved for. This
frequency is below resonance and the device is still of an inductive nature.
Data Sets
1 In the Model Builder under Results>Data Sets,
right-click Solution 4
and choose Duplicate .
2 Right-click Solution 5
Add Selection .
and choose 
3 Go to the Selection settings window. Under
Geometric Entity Selection from the Geometric entity
level list, choose Boundary.
4 From the Selection list, choose Conductor
Boundaries.
42 | Tutorial Example: Modeling a 3D Inductor
3D Plot Group 4
1 In the Model Builder, right-click Results
and choose 3D Plot Group . A 3D Plot
Group 4 node is added to the sequence.
2 Go to the 3D Plot Group settings window.
Under Data from the Data set list, choose
Solution 5.
3 From the Parameter value (freq) list, 
choose 10e5.
4 Right-click 3D Plot Group 4
choose Surface .
and 
5 Go to the Surface settings window. In the
upper-right corner of the Expression
section, click Replace Expression
.
6 From the menu, choose
Magnetic Fields>Currents and charge>
Surface current density norm (mf.normJs)
(when you know the variable name, as in
this exercise, you can also enter mf.normJs
directly in the Expression field).
7 Click the Plot
button.
Tutorial Example: Modeling a 3D Inductor
| 43
The plot shows the surface current distribution and should match this figure.
Change to the highest frequency solved for and compare the results. Finish the
modeling session by plotting the real and imaginary parts of the coil impedance.
1D Plot Group 5
1 In the Model Builder, right-click Results
and choose 1D Plot Group
.
2 In the 1D Plot Group settings window under Data, select Solution 4 from the
Data set list.
3 Under Results right-click 1D Plot Group 5
and choose Global
.
4 Go to the Global settings window. Under y-Axis Data in the Expression
column, enter real(mf.Zport_1). In the Description column, enter Lumped
port impedance.
to see what variables
Note: In general, you can click the Replace Expression
are available in the predefined lists. In this case, you can enter the information
directly into the table.
44 | Tutorial Example: Modeling a 3D Inductor
5 Click the Plot
button.
The resistive part of the coil impedance peaks at the resonance frequency.
1D Plot Group 6
1 In the Model Builder, right-click Results
and choose 1D Plot Group
.
2 Go to the 1D Plot Group settings window. Under Data from the Data set list,
select Solution 4.
3 Under Results right-click 1D Plot Group 6
and choose Global
.
4 Go to the Global settings window. Under y-Axis Data in the Expression
column, enter imag(mf.Zport_1). In the Description column, enter Lumped
port impedance.
Tutorial Example: Modeling a 3D Inductor
| 45
5 Click the Plot
button.
The reactive part of the coil impedance changes sign when passing through the resonance frequency,
going from inductive to capacitive.
As a final step, pick one of the plots to use as a model thumbnail.
1 In the Model Builder under Results click 3D Plot Group 2
.
2 From the File menu, choose Save Model Thumbnail.
To view the thumbnail image, click the Root node and look under the Model
Thumbnail section. Make adjustments to the image in the Graphics window using
the toolbar buttons until the image is one that is suitable to your purposes.
This concludes this introduction.
46 | Tutorial Example: Modeling a 3D Inductor